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Evgeny Shinder

Publications and source records attributed to Evgeny Shinder.

At least 19 recordsLinked to original sources
arXiv2026

Obstructions for codimension one multiple fibers of Lagrangian and Calabi--Yau fibrations

We prove that compact hyper-K\"ahler manifolds with a Lagrangian fibration over a projective space have no multiple fibers in codimension one. This has several consequences for the structure of Lagrangian fibrations, including progress on Sawon's conjecture on general singular fibers, Kamenova--Lu anti-hyperbolicity, and the extension of the N\'eron model action to a big open subset of the base. We prove the same result for Calabi--Yau fibrations on simply-connected K-trivial varieties, including elliptic fibrations, with a single exceptional case: an odd-dimensional K-trivial variety with a single fiber of multiplicity 2 over the projective line. This exceptional case is realized by examples of Borisov--Nuer and their generalizations.

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arXiv2025

Atomic decompositions for derived categories of G-surfaces

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

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arXiv2025

Unboundedness for motivic invariants of birational automorphisms

We introduce horizontal and vertical motivic invariants of birational maps between rational dominant maps and study their basic properties. As a first application, we show that the (usual) motivic invariants vanish for birational automorphisms of threefolds over algebraically closed fields of characteristic zero. On the other hand, we prove that the motivic invariants of the birational automorphism group of many types of varieties, including projective spaces of dimension at least four over a field of characteristic zero, do not form a bounded family, even after extending scalars to the algebraic closure of the field. For such varieties, we further show that their birational automorphism groups are not generated by maps preserving a conic bundle or a rational surface fibration structure, and their abelianizations do not stabilize.

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arXiv2024

Twists of intermediate Jacobian fibrations

We study the sections, Tate--Shafarevich twists, and the period for an OG10 hyperk\"ahler Lagrangian associated to a cubic fourfold. To do so, we introduce the analytic relative Jacobian sheaf for a Lagrangian fibration of a hyperk\"ahler variety. The Tate--Shafarevich group parameterizing twists is isomorphic to the first cohomology group of this sheaf and we compute it in terms of certain analytic Brauer groups associated to the cubic fourfold. We prove that the primitive Hodge lattice of the cubic fourfold is, up to a sign, isometric to a distinguished sublattice of the second cohomology group of the associated OG10 hyperk\"ahler manifold. Among the main tools we use are intersection complexes with integral coefficients, Decomposition Theorem, Hodge modules and Deligne cohomology.

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arXiv2024

Derived equivalence of elliptic K3 surfaces and Jacobians

We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.

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arXiv2023

Homological Bondal-Orlov localization conjecture for rational singularities

Given a resolution of rational singularities $π\colon \tilde{X} \to X$ over a field of characteristic zero we use a Hodge-theoretic argument to prove that the image of the functor $\mathbf{R}π_*\colon \mathbf{D}(\tilde{X}) \to \mathbf{D}(X)$ between bounded derived categories of coherent sheaves generates $\mathbf{D}(X)$ as a triangulated category. This gives a weak version of the Bondal-Orlov localization conjecture, answering a question of Pavic and Shinder. The same result is established more generally for proper (non-necessarily birational) morphisms $π\colon \tilde{X} \to X$, with $\tilde{X}$ smooth, satisfying $\mathbf{R}π_*(\mathcal{O}_{\tilde{X}}) = \mathcal{O}_X$.

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arXiv2023

Factorization centers in dimension two and the Grothendieck ring of varieties

We initiate the study of factorization centers of birational maps, and complete it for surfaces over a perfect field in this article. We prove that for every birational automorphism $ϕ: X \dashrightarrow X$ of a smooth projective surface $X$ over a perfect field $k$, the blowup centers are isomorphic to the blowdown centers in every weak factorization of $ϕ$. This implies that nontrivial L-equivalences of $0$-dimensional varieties cannot be constructed based on birational automorphisms of a surface. It also implies that rationality centers are well-defined for every rational surface $X$, namely there exists a $0$-dimensional variety intrinsic to $X$, which is blown up in any rationality construction of $X$.

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arXiv2023

Motivic invariants of birational maps

We construct invariants of birational maps with values in the Kontsevich--Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure of the Grothendieck ring and L-equivalence. Building on known constructions of L-equivalence, we prove new unexpected results about Cremona groups.

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arXiv2023

Derived categories of Fano threefolds and degenerations

Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree $d \in \{2,3,4,5\}$ and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree $d = 1$ can be smoothly deformed to the nontrivial components of the derived categories of prime Fano threefolds of genus $g = 2d + 2 \in \{4,6,8,10,12\}$. This corrects and proves the Fano threefolds conjecture of the first author from [Kuz09], and opens a way to interesting geometric applications, including a relation between the intermediate Jacobians and Hilbert schemes of curves of the above threefolds. We also describe a compactification of the moduli stack of prime Fano threefolds endowed with an appropriate exceptional bundle and its boundary component that corresponds to degenerations associated with del Pezzo threefolds.

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arXiv2023

On automorphic forms of small weight for fake projective planes

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least $9$ automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The first proof is based on the observation that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, \emph{non-abelian}, Heisenberg group. For the second proof, we also demonstrate vanishing of odd Betti numbers for a class of abelian covers, and use linearization of an auxiliary line bundle as well.

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arXiv2022

Homologically finite-dimensional objects in triangulated categories

In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for triangulated categories, and discuss crepant categorical contractions. We illustrate the introduced notions on examples of categories of geometric and algebraic origin and provide geometric applications. In particular, we apply our results to prove a bijection between semiorthogonal decompositions of the derived category of a singular variety and the derived category of its smoothing with support on the central fiber.

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arXiv2022

Categorical absorptions of singularities and degenerations

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$.

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arXiv2021

Derived categories of nodal del Pezzo threefolds

We give a complete answer for the existence of Kawamata type semiorthogonal decompositions of derived categories of nodal del Pezzo threefolds. More precisely, we show that nodal del Pezzo threefolds of degree $1\leq d \leq 4$ have no Kawamata type decomposition and that all nodal del Pezzo threefolds of degree $5$ admit a Kawamata decomposition. For the proof we go through the classification of singular del Pezzo threefolds, compute divisor class groups of nodal del Pezzo threefolds of small degree and use projection from a line to construct Kawamata semiorthogonal decompositions for the degree $5$ case. An analogous decomposition of the nodal del Pezzo threefold of degree $6$ has been recently constructed by Kawamata. Our construction of the Kawamata decomposition for a singular del Pezzo threefold of degree $5$ fits into a family of semiorthogonal decompositions (which we call a relative tilting decomposition) interpolating between a Kawamata decomposition on a singular fiber and a full exceptional collection on the smooth fibers.

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arXiv2020

Obstructions to semiorthogonal decompositions for singular threefolds I: K-theory

We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic $\mathrm{K}$-theory and translate them into the concept of maximal nonfactoriality. Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees $1 \le d \le 4$ with maximal class group rank. We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components.

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arXiv2020

Derived categories of singular surfaces

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface $X$ with rational singularities from a semiorthogonal decomposition of its resolution. In the case when $X$ has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of $X$. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1.

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arXiv2019

Variation of Stable Birational Types of Hypersurfaces

We introduce and study the question how can stable birational types vary in a smooth proper family. Our starting point is the specialization for stable birational types of Nicaise and the author and our emphasis is on stable birational types of hypersurfaces. Building up on the work of Totaro and Schreieder on stable irrationality of hypersurfaces of high degree, we show that smooth Fano hypersurfaces of large degree over a field of characteristic zero are in general not stably birational to each other. In the appendix Claire Voisin proves a similar result in a different setting using the Chow decomposition of diagonal and unramified cohomology.

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arXiv2019

L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr(2,5), and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg--Shafarevich theory of twisting for elliptic surfaces. Finally, we apply our results to K3 surfaces and investigate when the two elliptic L-equivalent K3 surfaces we construct are isomorphic, using Neron--Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L-equivalence is new.

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